A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing Erik Meijering Proceedings of the IEEE, vol. 90, no. 3, March 2002, pp. 319–342 Abstract—This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolution-based interpolation, history, image processing, polyno- mial interpolation, signal processing, splines. It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it.1 I Introduction The problem of constructing a continuously-defined function from given discrete data is unavoidable whenever one wishes to manipulate the data in a way that requires information not included explicitly in the data. In this age of ever increasing digitization in the storage, processing, analysis, and communication of information, it is not difficult to find examples of applications where this problem occurs. The relatively easiest and in many applications often most desired approach to solve the problem is interpolation, where an approximating function is constructed in such a way as to agree perfectly with the usually unknown original function at the given measurement points.2 In view of its increasing relevance, it is only natural that the subject of interpolation is receiving more and more attention these days.3 However, in times where all efforts are directed towards the future, the past may 1Leibniz, in the opening paragraph of his Historia et Origo Calculi Differentialis [188]. The translation given here was taken from a paper by Child [49]. 2The word “interpolation” originates from the Latin verb interpolare, a contraction of “inter”, meaning “between”, and “polare”, meaning “to polish”. That is to say, to smooth in between given pieces of information. It seems that the word was introduced in the English literature for the first time around 1612, and was then used in the sense of “to alter or enlarge [texts] by insertion of new matter” [293]. The original Latin word appears [10] to have been used first in a mathematical sense by Wallis in his 1655 book on infinitesimal arithmetic [341]. 3A search in the multidisciplinary databases of bibliographic information collected by the Institute for Scientific Information in the Web of Science will reveal that the number of publications containing the word “interpolation” in the title, list of keywords, or the abstract, has dramatically increased over the past decade, even when taking into account the intrinsic (and likewise dramatic) increase in the number of publications as a function of time. PP-2 A Chronology of Interpolation easily be forgotten. It is no sinecure, scanning the literature, to get a clear picture of the development of the subject through the ages. This is quite unfortunate, since it implies a risk of researchers going over grounds covered earlier by others. History has shown many examples of this, and several new examples are revealed here. The goal of the present paper is to provide a systematic overview of the developments in interpolation theory, from the earliest times to the present date, and to put the most well-known techniques currently used in signal and image processing applications into historical perspective. The paper is intended to serve as a tutorial and a useful source of links to the appropriate literature for anyone interested in interpolation, whether it be its history, theory, or applications. As already suggested by the title, the organization of the paper is largely chronological. Section II presents an overview of the earliest-known uses of interpolation in antiquity and describes the more sophisticated interpolation methods developed in different parts of the world during the Middle Ages. Next, Section III discusses the origins of the most important techniques developed in Europe during the period of Scientific Revolution, which in the present context lasted from the early 17th until the late 19th century. A discussion of the developments in what could be called the Information and Communication Era, covering roughly the past century, is provided in Section IV. Here, the focus of attention is on the results that have had the largest impact on the advancement of the subject in signal and image processing, in particular on the development of techniques for the manipulation of intensity data defined on uniform grids. Although recently developed alternative methods for specific interpolation tasks in this area are also mentioned briefly, the discussion in this part of the paper is restricted mainly to convolution-based methods, which is justified by the fact that these are the most frequently used interpolation methods, probably because of their versatility and relatively low complexity. Finally, summarizing and concluding remarks are made in Section V. II Ancient Times and the Middle Ages In his 1909 book on interpolation [316], Thiele characterized the subject as “the art of reading between the lines in a [numerical] table”. Examples of fields in which this problem arises naturally and inevitably are astronomy and, related to this, calendar computation. And because man has been interested in these since day one, it should not surprise us that it is in these fields that the first interpolation methods were conceived. This section discusses the earliest-known contributions to interpolation theory. II.A Interpolation in Ancient Babylon and Greece In antiquity, astronomy was all about time keeping and making predictions concerning astronomical events. This served important practical needs: farmers, for example, would base their planting strategies on these predictions. To this end it was of great importance to keep up lists—so-called ephemerides—of the positions of the sun, moon, and the known planets for regular time intervals. Obviously these lists would contain gaps, due to either atmospherical conditions hampering observation or the fact that celestial bodies may not be visible during certain periods. From his study of ephemerides found on ancient astro- nomical cuneiform tablets originating from Uruk and Babylon in the Seleucid period (the last three centuries BC), the historian-mathematician Neugebauer [230, 231] concluded that interpolation was used in order to fill these gaps. Apart from linear interpolation, the tablets also revealed the use of more complex interpolation methods. Precise formulations of the latter methods have not survived, however. II Ancient Times and the Middle Ages PP-3 An early example of the use of interpolation methods in ancient Greece dates from about the same period. Toomer [318] believes that Hipparchus of Rhodes (190–120 BC) used linear interpolation in the construction of tables of the so-called “chord function” (re- lated to the sine function) for the purpose of computing the positions of celestial bodies. Later examples are found in the Almagest (“The Mathematical Compilation”, ca. 140 AD) of Claudius Ptolemy, the Egypt-born Greek astronomer-mathematician who propounded the geocentric view of the universe which prevailed until the 16th century. Apart from theory, this influential work also contains numerical tables of a wide variety of trigonomet- ric functions defined for astronomical purposes. To avoid the tedious calculations involved in the construction of tables of functions of more than one variable, Ptolemy used an ap- proach that amounts to tabulating the function only for the variable for which the function varies most, given two bounding values of the other variable, and to provide a table of coefficients to be used in an “adaptive” linear interpolation scheme for computation of the function for intermediate values of this latter variable [335]. II.B Interpolation in Early-Medieval China and India Analysis of the computational techniques on which early-medieval Chinese ephemerides are based often reveals the use of higher-order interpolation formulae.4 The first person to use second-order interpolation for computing the positions of the sun and the moon in constructing a calendar is said to be the astronomer Li`uZhu´o. Around 600 AD he used this technique in producing the so-called Hu´ang j´ıl`ı, or “Imperial Standard Calendar”. AccordingtoL˘ıY˘an & D`uSh´ır´an [204], the formula involved in his computations reads, in modern notation:5 ξ ξ2 f(x0 + ξT)=f(x0)+ (∆1 +∆2)+ξ(∆1 − ∆2) − (∆1 − ∆2), (1) 2 2 with 0 6 ξ<1, T>0, ∆1 = f(x0 +T )−f(x0), and ∆2 = f(x0 +2T )−f(x0 +T ), and with f(x0),f(x0 + T ), and f(x0 +2T ) the observed results at times x0,x0 + T ,andx0 +2T , respectively. This formula is closely related to later Western interpolation formulae, to be discussed in the next section. Methods for second-order interpolation of unequal-interval observations were later used by the astronomer Monk Y`ıX´ıng in producing the so-called “D`aY˘an Calendar” (727 AD) and by X´u Ang´ in producing the “Xu¯an M´ıng Calendar” (822 AD). The latter also used a second-order formula for interpolation of equal-interval observations equivalent to the formula used by Li`uZhu´o. Accurate computation of the motion of celestial bodies, however, requires more sophis- ticated interpolation techniques than just second order. More complex techniques were later developed by Gu¯o Sh˘ouj`ıng and others. In 1280 AD they produced the so-called Sh`ou sh´ıl`ı, or “Works and Days Calendar”, for which they used third-order interpolation.